Calvin Webber (Christopher Walken) is a brilliant but paranoid scientist and inventor in 1963 suburban Southern California. Fearing the threat of the Soviet Union, he builds a huge fallout shelter in his back yard. When the Cuban missile crisis is announced on television, he and his pregnant wife, Helen (Sissy Spacek), go into the underground shelter. Coincidentally, a jet aircraft crashes on their house - hearing the noise convinces them that the bomb has been dropped, and they plan for long stay, until the air is clear enough of nuclear fallout to emerge.

The bomb shelter is a massive reconstruction of their 60s lifestyle, and includes not only food and water for 35 years, but all the comforts of home. In a few weeks, Helen gives birth to a son and, because of the circumstances of his arrival, they name him Adam. As the boy grows,his father tutors him on many subjects, science, literature, geography, etc., and his mother teaches him social etiquette and dancing.

A restaurant is built on the property where their destroyed home sat and, over the ensuing years, it goes through a variety of styles and clientele. The surrounding residential area becomes more urbanized and eventually degenerates into a seedy skid row.

After the 35 years have passed, Calvin Webber opens the sealed door and takes the elevator up through the floor of the now abandoned restaurant. His protective suit and gas causes the lone inhabitant (Joey Slotnick) to believe that aliens have arrived. After encountering degenerates, panhandlers and prostitutes, Calvin concludes that post-bomb life on Earth is made up of mutants, and he returns to the shelter.

But needing supplies, he prepares to send Adam (Brenden Fraser) above, with $3000 in cash and a long shopping list. Adam also takes his baseball card collection and a number of old stock certificates as possible additional revenue sources.

Adam is also aghast at the inhabitants he encounters and they are equally perplexed with his mix of naive innocence and impeccable manners. He finds a sports card shop and the proprietor offers to buy his entire baseball card collection for $500, but his disgusted assistant, Eve Rustikoff (Alicia Silverstone) informs him that the Mantle rookie card alone is worth thousands. Her boss fires her and Adam, feeling guilty, offers to help her. He hires her to help sell his cards and to buy truckloads of supplies, without revealing the purpose.

Eve, a victim of too many failed romances, keeps an emotional distance between herself and Adam but, nevertheless, is fascinated by him. She and her gay friend Troy (Dave Foley) take Adam to a nightclub where, to their amazement and to the delight of the patrons, he puts on a fabulous swing dancing performance.

Adam finally confides to Eve where he has come from and his purpose in obtaining the supplies, but she concludes that he is mentally ill and calls Social Services. But he escapes their attempts to take him away, and returns to the shelter. Eve and Troy find the1960s stock certificates he left behind and determine that they are worth millions and millions of dollars. That and a tube of Ipana toothpaste convince them that he was telling the truth. They desperately search for him, knowing only that the bomb shelter was in the vicinity of an adult book store. She finds Adam on the street and they embrace.

After Eve meets Adam's parents in the shelter, they have a house built above for them, in the same style as the 1960s home they left, so as to gradually reintroduce them to the world. When Adam's father is informed that there was no bomb, he laughs it off as a Soviet trick and immediately starts pacing off an area for his next underground shelter.